Separability Properties of Monadically Dependent Graph Classes

TL;DR

This paper characterizes monadically dependent graph classes through flip-separability, showing that bounded local modifications can control the distribution of weights within graph neighborhoods.

Contribution

It introduces flip-separability as a precise characterization of monadically dependent classes and develops a new toolbox for local separation techniques.

Findings

Monadically dependent classes are exactly flip-separable.

Flip operations can control local weight distributions in graphs.

Provides a new framework for analyzing local separations in graph classes.

Abstract

A graph class $\mathcal C$ is monadically dependent if one cannot interpret all graphs in colored graphs from $\mathcal C$ using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by the following property, which we call flip-separability: for every $r\in \mathbb{N}$, $\varepsilon>0$, and every graph $G\in \mathcal{C}$ equipped with a weight function on vertices, one can apply a bounded (in terms of $\mathcal{C},r,\varepsilon$) number of flips (complementations of the adjacency relation on a subset of vertices) to $G$ so that in the resulting graph, every radius-$r$ ball contains at most an $\varepsilon$-fraction of the total weight. On the way to this result, we introduce a robust toolbox for working with various notions of local separations in monadically dependent classes.